Find the Function f(x)=-5x+3

$f (x) = - 5 x + 3$

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Split the single integral into multiple integrals.

$\int - 5 x d x + \int 3 d x$

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$- 5 \int x d x + \int 3 d x$

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 5 (\frac{1}{2} x^{2} + C) + \int 3 d x$

Apply the constant rule.

$- 5 (\frac{1}{2} x^{2} + C) + 3 x + C$

Simplify.

Combine $\frac{1}{2}$ and $x^{2}$ .

$- 5 (\frac{x^{2}}{2} + C) + 3 x + C$

Simplify.

$\frac{- 5 x^{2}}{2} + 3 x + C$

Move the negative in front of the fraction.

$- \frac{5 x^{2}}{2} + 3 x + C$

Reorder terms.

$- \frac{5}{2} x^{2} + 3 x + C$

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$F (x) = - \frac{5}{2} x^{2} + 3 x + C$